Table Of ContentTRANSVERSALITY OF HOLOMORPHIC MAPPINGS BETWEEN
REAL HYPERSURFACES IN DIFFERENT DIMENSIONS
7
0
0
M.S. BAOUENDI, PETER EBENFELT, AND LINDA P. ROTHSCHILD
2
n
a Abstract. Inthispaper,weconsiderholomorphicmappingsbetweenrealhypersurfaces
J indifferentdimensionalcomplexspaces. Wegiveanumberofconditionsthatimplythat
6 such mappings are transversalto the target hypersurface at most points.
1
]
V
1. Introduction and Main Results
C
.
h Inthispaper,weconsider holomorphicmappingsbetweenrealhypersurfaces indifferent
t
a dimensional complexspaces. Wegiveanumber ofconditionsimplyingthatsuchmappings
m aretransversaltothetargethypersurfaceatmostpoints. RecallthatifU isanopensubset
[ of Cn+1, H a holomorphic mapping U Cn′+1, and M′ a real hypersurface through a
1 point H(p) for some p U, then H is sa→id to be transversal to M′ at H(p) if
v ∈
2 (1.1) T M′ +dH(T Cn+1) = T Cn′+1,
H(p) p H(p)
3
4 where TpCn+1 and TH(p)M′ denote the real tangent spaces of Cn+1 and M′ at p and H(p),
1 respectively. (Wemention thatthenotionoftransversality of amapping toa hypersurface
0
coincides with that of CR transversality; cf. [ER06].) We shall assume that there is a real
7
0 hypersurface M U such that H(M) M′. Then transversality at a point H(p), for
/ ⊂ ⊂
h p M, is equivalent to the nonvanishing at p of the normal derivative of the real function
∈
t u := ρ′ H, where ρ′ = 0 is a local defining equation for M′ near H(p). Hence a result
a
◦
m on transversality can be regarded as a type of Hopf Lemma.
: The equidimensional case (i.e. n = n′) has been considered by many authors; we men-
v
i tion here the papers [For76], [For78], [Pin77], [BB82], [BR90], [BR93], [CR94], [CR98],
X
[BHR95], [HP96], [ER06], [LM06]. In the equidimensional case, transversality holds at a
r
a point H(p) under rather general conditions. For instance, in [ER06] it is proved that H
is transversal to M′ at H(p) provided that M′ is of finite type at H(p) and the generic
rank of H , where Σ denotes the Segre variety of M at p, is n. The situation in the
|Σp p
case where n′ > n is much more complicated. Indeed, transversality may fail at a point
H(p) even for a polynomial embedding C2 C3 sending one nondegenerate hyperquadric
→
into another, as is illustrated by Example 2.4 below. Observe that a trivial case where
The firstandthirdauthorsaresupportedinpartby theNSF grantDMS-0400880. The secondauthor
is supported in part by the NSF grant DMS-0401215.
2000 Mathematics Subject Classification. 32H35, 32V40.
1
2 M.S. BAOUENDI,P. EBENFELT, ANDL. P. ROTHSCHILD
transversality fails at all points is when H(U) is contained in M′. In the equidimensional
case, this is the only way for transversality to fail at all points provided that M is holo-
morphically nondegenerate (see Example 2.2 and Theorem 5.1). When n′ > n, this is no
longer the case, as is illustrated by Theorem 1.4 as well as Example 2.5. Our Theorems
1.1 and 1.3 give conditions that guarantee transversality at most points. The results are
essentially optimal, as is illustrated by examples. Having tranversality at most points
is crucial in the study of rigidity of embeddings into hyperquadrics. See e.g. [Web79],
[Far86], [CS83], [For86], [D’A88], [Hua99], [EHZ04], [EHZ05], [BH05]. See also [Z07] and
[A07] for recent related work on transversality of holomorphic Segre mappings.
Before stating our main results, we introduce some notation. Let M be a hypersurface
in Cn+1, p M, and : Cn Cn C a representative of the Levi form of M at p.
∈ L × →
We shall denote by e(M,p) := min(e ,e ) and e (M,p) = e , where e ,e ,e , denote
− + 0 0 + − 0
the number of positive, negative, and zero eigenvalues of at p. Observe that e(M,p)
L
and e (M,p) are independent of the choice of representative of the Levi form. A
0
L
connected hypersurface M is said to be holomorphically nondegenerate if there are no
germs of nontrivial holomorphic (1,0)-vector fields tangent to M. We point out that if
M is connected and Levi nondegenerate at some point, i.e. e (M,p) = 0 for some p M,
0
∈
then M is necessarily holomorphically nondegenerate. The converse is not true. The
reader is referred to [BER99] for further details on this and other related notions (see also
[Sta95] for holomorphic nondegeneracy).
In our first theorem, we give two independent conditions guaranteeing transversality at
most points.
Theorem 1.1. Let M Cn+1, and M′ Cn′+1 be connected real-analytic hypersur-
faces and U an open ne⊂ighborhood of M i⊂n Cn+1. Assume that M is holomorphically
nondegenerate and that either
(1.2) e(M′,p′)+e (M′,p′) n 1, p′ M′
0
≤ − ∀ ∈
or
(1.3) n′ +e (M′,p′) 2n, p′ M′,
0
≤ ∀ ∈
holds. If H : U Cn′+1 is a holomorphic mapping with H(M) M′, then one of the
→ ⊂
following two mutually exclusive conditions holds.
(i) There is an open subset V U with M V and H(V) M′.
⊂ ⊂ ⊂
(ii) H is transversal to M′ at H(p) for all p M outside a proper real-analytic subset.
∈
Remark 1.2. We point out that (i) holds if and only if there exists a point p M and a
open neighborhood W U of p in Cn+1 such that H(W) M′. This follows e∈asily from
⊂ ⊂
the connectedness andreal-analyticity of M. Similarly, (ii) holds if and only if there exists
p M such that H is transversal to M′ at H(p). Indeed, if H is transversal at p M,
∈ ∈
then we have ρ′ H = aρ, where ρ and ρ′ are local real-analytic defining functions near p
◦
and H(p) respectively and a is a real-analytic function defined near p, with a(p) = 0. The
6
TRANSVERSALITY OF HOLOMORPHIC MAPPINGS BETWEEN REAL HYPERSURFACES 3
set of points, near p, at which H is not transversal is given by the equation a = 0, which
defines a proper real-analytic subset of M near p. A standard connectedness argument
shows that (ii) holds.
Thecondition(1.2)inTheorem1.1isoptimal, ascanbeseenbyExample2.5. Similarly,
Theorem 1.4 below shows that condition (1.3) is also optimal. However, if M and M′ are
Levi nondegenerate and the target is a hyperquadric1, then condition (1.3) in Theorem
1.1 can be weakened, as is shown by the following result.
Theorem 1.3. Let M Cn+1 be a connected, real-analytic hypersurface and U an open
neighborhood of M in C⊂n+1. Let M′ Cn′+1 be a nondegenerate hyperquadric. Suppose
that n′ 3(n e (M,p)) for some p⊂oint p M. If H : U Cn′+1 is a holomorphic
0
≤ − ∈ →
mapping with H(M) M′ , then one of the following mutually exclusive conditions must
⊂
hold.
(i) There is an open subset V U with M V and H(V) M′.
⊂ ⊂ ⊂
(ii) H is transversal to M′ at H(p) for all p M outside a proper real-analytic subset.
∈
Weshouldremarkthatconclusion(ii)inTheorems1.1and1.3cannotbereplacedbythe
stronger conclusion thattransversality holdsforeveryp M, asisshown byExamples 2.3
∈
and 2.4. In the equidimensional case, condition (1.3) is always satisfied. The conclusion
of Theorem 1.1, in this case, can be deduced from known results (e.g. [BR90] and [ER06])
by using also Theorem 5.1 of the present paper. Even in the equidimensional case, the
conclusion (ii) cannot be replaced by that of transversality for all p M as is shown by
∈
Example 6.2 in [ER06]. However, if the condition that M is of finite type is added, then
it is unknown if this replacement can be made (see Conjecture 2.7 in [LM06]; see also
Question 1 in [ER06]).
The following result shows that the condition in Theorem 1.3 requiring M′ to be a
nondegenerate hyperquadric cannot be replaced by the weaker assumption that M′ is a
Levi nondegenerate hypersurface. As mentioned above, it also shows that the condition
(1.3) in Theorem 1.1 is optimal.
Theorem 1.4. Given M Cn+1 a nondegenerate hyperquadric, there exist a Levi non-
degenerate hypersurface M⊂′ C2n+2 and H : Cn+1 C2n+2 a polynomial embedding of
⊂ →
degree 2 such that H sends M into M′, but neither (i) nor (ii) of Theorem 1.1 holds.
More precisely, if M := Z Cn+1: ρ(Z,Z¯) = 0 with
{ ∈ }
n
(1.4) ρ(Z,Z¯) := Im w δ z 2, Z = (z,w) Cn C,
j j
− | | ∈ ×
j=1
X
where δ = 1 and , is a nondegenerate hermitian form in C2n+1 with n negative
j
± h· ·i
and n + 1 positive eigenvalues, then there exist a polynomial embedding of degree two,
1By a hyperquadric in Cn+1, we mean a real-algebraic hypersurface defined by Im w = z,z¯ , where
, is a hermitian form in Cn. h i
h· ·i
4 M.S. BAOUENDI,P. EBENFELT, ANDL. P. ROTHSCHILD
H : Cn+1 C2n+2, and a real bihomogeneous polynomial φ(z′,z¯′), z′ C2n+1, of bidegree
→ ∈
(2,2), such that if
(1.5) ρ′(z′,w′,z¯,w¯) := Im w′ z′,z¯′ φ(z′,z¯′)
−h i−
then ρ′ H = 4ρ2.
◦ −
We should point out that if the target hypersurface M′ in either Theorem 1.1 or 1.3
does not contain any nontrivial complex subvarieties, then condition (i) is equivalent to
the mapping H being constant. Hence, if the hypothesis that M′ does not contain any
nontrivial complex subvarieties is added to either Theorem 1.1 or 1.3 and H is assumed
to be nonconstant, then the conclusion (ii) necessarily follows. In the last section of this
paper, we give a number of sufficient conditions for (i) to hold (see Theorems 5.1, 5.7 and
corollaries). In the equidimensional case (i.e. n = n′), we give two conditions equivalent
to (i) (see Corollary 5.2).
2. Examples and a lemma
In this section, we give some examples, which show that our main results are sharp.
We begin with the following lemma, which expresses conditions (i) and (ii) in Theorems
1.1 and 1.3 in terms of local defining functions for M and M′.
Lemma 2.1. Let M Cn+1 and M′ Cn′+1 be connected real-analytic hypersurfaces and
U an open neighborho⊂od of M in Cn+⊂1. Let p M and p′ M′ and suppose that M and
M′ are defined locally by ρ = 0 and ρ′ = 0 near∈p and p′, re∈spectively. Let H: U Cn′+1
→
be a holomorphic mapping with H(M) M′ and H(p) = p′. If (i) in Theorem 1.1 does
⊂
not hold, then there exists a unique integer k 1 such that ρ′ H = aρk, where a is a
real-valued, real-analytic function defined near≥p in Cn+1 with ◦a 0. Moreover, the
M
| 6≡
condition (ii) in Theorem 1.1 is equivalent to k = 1.
Proof. Since H(M) M′ and(i) does not hold, ρ′ H vanishes onM but is not identically
⊂ ◦
zero near p (by Remark 1.2). Hence, ρ′ H = bρ, where b 0. By unique factorization,
◦ 6≡
there is a unique integer l 0 such that b = aρl with a 0. Now, (ii) is equivalent to
M
≥ | 6≡
b 0, in view of Remark 1.2, and hence k = 1+l = 1. The conclusion of the lemma
M
| 6≡ (cid:3)
now follows.
Example 2.2. LetM C2 betheLevi-flathypersurface givenbyρ(z,w,z,w) := Im w =
0, and M′ C2 the h⊂ypersurface given by ρ′(z′,w′,z′,w′) := Im w′ z′ 2 = 0. The
mapping H⊂: C2 C2 given by H(z,w) = (w,iw2) maps M into−M|′,|but satisfies
→
neither (i) nor (ii) of Theorem 1.1. Indeed, we have ρ′ H 2ρ2. Note that M is not
◦ ≡ −
holomorphically nondegenerate, which is the only assumption of Theorem 1.1 in this case
(n′ = n = 1).
Example 2.3. Let M C3 be the unit sphere,
⊂
ρ(Z,Z¯) := Z 2 + Z 2 + Z 2 1 = 0,
1 2 3
| | | | | | −
TRANSVERSALITY OF HOLOMORPHIC MAPPINGS BETWEEN REAL HYPERSURFACES 5
and M′ C5 be the hyperquadric defined by
⊂
ρ′(z,′,w′,z¯′,w¯′) := Im w′ ( z′ 2 + z′ 2 + z′ 2 z′ 2) = 0.
− | 1| | 2| | 3| −| 4|
Consider the mapping
H(Z) := (Z Z ,Z2,Z Z ,Z ,0).
1 2 2 2 3 2
Observe that we have the identity ρ′(H(Z),H(Z)) = Z 2ρ(Z,Z¯). We conclude that
2
H sends M into M′, H(C3) is not contained in M′, a−nd| H| is not transversal to M′ at
0. Note also that Theorem 1.1 applies, since condition (1.2) holds. This example shows
that, under assumption (1.2), conclusion (ii) in Theorem 1.1 cannot be replaced by the
stronger conclusion of transversality at all points of M.
Example 2.4. Let M C2 be the hypersurface given by ρ(z,w,z,w) := Im w z 2 = 0,
M′ C3 the Levi-nond⊂egenerate hyperquadric given by ρ′(z′,w′,z′,w′) := Im w−′+| |z′ 2
z′ 2⊂= 0 and H : C2 C3 given by | 1| −
| 2| →
i i
H(z,w) = (z +z2 + w,z z2 w, 2zw).
2 − − 2 −
We have ρ′ H 2(z + z)ρ. Hence H is transversal on M outside the real-analytic
◦ ≡ −
submanifold of M given by Re z = 0. For every p′ M′, we have e (M′,p′) = 0
0
∈
and e(M′,p′) = 1. Hence, (1.3) in Theorem 1.1 holds (but (1.2) does not). Also, the
assumption on n′ in Theorem 1.3 holds. Moreover, M is holomorphically nondegenerate,
(i) does not hold, and transversality does not hold at every point of M. This example
shows that (ii) in Theorems 1.1 and 1.3 cannot be replaced by the stronger condition
of transversality at all points of M. Observe that H(C2) is the 2-dimensional complex
manifold given by (z′ +z′)(z′ z′)+iw′ = (z′ +z′)3/2.
1 2 1 − 2 1 2
Example 2.5. Let M C2 be the hypersurface given by ρ(z,w,z,w) := Im w z 2 = 0,
M′ C5 the Levi-nond⊂egenerate hyperquadric given by −| |
⊂
ρ′(z′,w′,z′,w′) := Im w′+ z′ 2 z′ 2 z′ 2 z′ 2 = 0,
| 1| −| 2| −| 3| −| 4|
and H : C2 C5 given by
→
H(z,w) = (iz +zw, iz +zw,w,√2z2,iw2).
−
We have ρ′ H 2ρ2. Since neither (i) nor (ii) ofTheorem 1.3holds, this example shows
◦ ≡ −
that the condition n′ 3(n e (M,p)) cannot be replaced by n′ 3(n e (M,p))+1.
0 0
≤ − ≤ −
3. Proof of Theorem 1.1
For the proof of Theorem 1.1 stated in the introduction, we shall need a number of
preliminary results, which may be of independent interest. Recall that if M is a real-
analytic hypersurface in Cn+1, defined locally near p M by the real-analytic equation
0
ρ(Z,Z¯) = 0, then the Segre variety of M at p, suffic∈iently close to p , is given by the
0
holomorphic equation ρ(Z,p¯) = 0. We shall denote the Segre variety of M at p by Σ .
p
The following proposition will be useful in the proofs of the main results.
6 M.S. BAOUENDI,P. EBENFELT, ANDL. P. ROTHSCHILD
Proposition 3.1. Let M Cn+1, and M′ Cn′+1 be connected real-analytic hypersur-
faces and U an open neigh⊂borhood of M in⊂Cn+1. If H : U Cn′+1 is a holomorphic
→
mapping with H(M) M′ and M is holomorphically nondegenerate, then at least one of
⊂
the following holds.
(i) There is an open subset V U with M V and H(V) M′.
⊂ ⊂ ⊂
(iii) For every p M outside a proper real-analytic subset, the rank of H at p is n.
∈ |Σp
Remark 3.2. We note that if, for some point p M, the restriction of H to the Segre
∈
variety of M at p has rank n at p, then (iii) in Proposition 3.1 holds. (This is true even
without assuming that M is holomorphically nondegenerate.) Indeed, the rank at p of
H is the rank of the n (n′ + 1) matrix given by (L H (p)), where j = 1,...,n,
|Σp × j k
k = 1,...,n′ + 1 and L ,...,L is a real-analytic local basis of the (1,0)-vector fields
1 n
tangent to M. Thus, if the rank of this matrix is n at some point, then it is n outside a
proper real-analytic subset of M near p. A standard connectedness argument shows that
(iii) holds.
Proof of Proposition 3.1. Weassume, inordertoreachacontradiction,thatneither(i)nor
(iii)ofProposition3.1holds. Letp M beapointatwhich M isfinitelynondegenerate2,
0
∈
andρ,ρ′ localdefiningfunctionsforM andM′ nearp andH(p ),respectively. ByLemma
0 0
2.1, there exists an integer k 1 such that
≥
(3.1) ρ′ H = aρk,
◦
where a is not identically zero on M. By moving to a nearby point, if necessary, we
may assume that a(p ) = 0. We choose normal coordinates (z,w) Cn C and
0
(z′,w′) Cn′ C for M6 and M′ vanishing at p and H(p ), respective∈ly. H×ence, the
0 0
∈ ×
defining equations of M and M′ can be written as w = Q(z,z¯,w¯) and w′ = Q′(z′,z¯′,w¯′),
respectively, with Q(z,0,τ) Q(0,χ,τ) Q′(z′,0,τ) Q′(0,χ′,τ) τ. We write
≡ ≡ ≡ ≡
H(z,w) = (f(z,w),g(z,w)) with f = (f1,...,fn′). It follows from (3.1) that
(3.2) g(z,w) Q′(f(z,w),f¯(χ,τ),g¯(χ,τ)) = a(z,w,χ,τ)(w Q(z,χ,τ))k,
− −
with a(0) = 0. Setting χ = 0, τ = 0, we have g(z,w) = a(z,w,0,0)wk and hence
6
(3.3) g (0) = 0.
wk
6
2Recallthatareal-analytichypersurfaceM Cn+1 locallydefinednearapointp0 M byρ(Z,Z¯)=0
isfinitely nondegenerateatp0 ifthevectorsLα⊂ρZ(p0),α Zn+,spanallofCn+1,wher∈eρZ isthegradient
vectorofρ withrespecttoZ andLα =Lα11...Lαnn,with∈L1,...,Ln asinRemark3.2. IfM isconnected
and holomorphically nondegenerate, then it is finitely nondegenerate on a dense and open subset of M
(see [BHR96] and [BER99]).
TRANSVERSALITY OF HOLOMORPHIC MAPPINGS BETWEEN REAL HYPERSURFACES 7
We differentiate (3.2) k 1 times with respect to w and then set τ = 0, w = Q(z,χ,0).
−
We obtain, since g(z,0) 0,
≡
(3.4) gwk−1(z,Q(z,χ,0)) =
Q′(z′)α(f(z,Q(z,χ,0),f¯(χ,0),0)Pα(fw(z,Q(z,χ,0),...,fwk−1(z,Q(z,χ,0))),
|α|≤k−1
X
where the P (t ,...,t ) are universal polynomials. We now differentiate (3.4) with
α 1 k−1
respect to χ , for 1 j n, and then set χ = 0 to obtain
j
≤ ≤
(3.5) g (z,0)Q (z,0,0) =
wk χj
Q′(z′)αχ′(f(z,Q(z,χ,0),f¯(χ,0),0)f¯χj(0)Pα(fw(z,Q(z,χ,0),...,fwk−1(z,Q(z,χ,0))).
|α|≤k−1
X
Since (iii) does not hold, there exist constants a ,...,a with (a ,...,a ) = 0 such that
1 n 1 n
6
n
¯
(3.6) f (0)a = 0.
χj j
j=1
X
Thus, if we multiply (3.5) by a and sum over j, then we obtain
j
n
(3.7) g (z,0) a Q (z,0,0) 0.
wk j χj ≡
j=1
X
n n
It follows from (3.3) that a Q (z,0,0) 0 and, hence, a Q (0) = 0 for all
j=1 j χj ≡ j=1 j χjzα
multi-indices α. This contradicts the finite nondegeneracy of M at p and completes the
0
P P (cid:3)
proof of Proposition 3.1.
We mention here that some of the techniques in the proof of Proposition 3.1 were used
in [BH05]. The following transversality result may already be known in the folklore. For
the reader’s convenience, we include a proof here.
Proposition 3.3. Let M Cn+1, and M′ Cn′+1 be real-analytic hypersurfaces with
p M and p′ M′. Let ⊂H: (Cn+1,p) (⊂Cn′+1,p′) be a germ at p of a holomorphic
∈ ∈ →
mapping sending M into M′ and such that the restriction of H to the Segre variety of M
at p has rank n at p. If
(3.8) e(M′,p′)+e (M′,p′) n 1,
0
≤ −
then either H sends a full neighborhood of p in Cn+1 into M′ or H is transversal to M′
at p′.
Proof of Proposition 3.3. We assume, in order to reach a contradiction, that H does not
map an open neighborhood of p in Cn+1 into M′ and that H is not transversal to M′
at p′. We choose normal coordinates (z,w) Cn C and (z′,w′) Cn′ C for M and
∈ × ∈ ×
M′, vanishing at p and p′, respectively. We write H(z,w) = (f(z,w),g(z,w)) with f =
8 M.S. BAOUENDI,P. EBENFELT, ANDL. P. ROTHSCHILD
(f1,...,fn′). The defining equations of M and M′ can be written as w = Q(z,z¯,w¯) and
w′ = Q′(z′,z¯′,w¯′), respectively, with Q(z,0,τ) Q(0,χ,τ) Q′(z′,0,τ) Q′(0,χ′,τ)
≡ ≡ ≡ ≡
τ. The fact that H maps M into M′ implies that
(3.9) g(z,w) Q′(f(z,w),f¯(χ,τ),g¯(χ,τ)) = a(z,w,χ,τ)(w Q(z,χ,τ)),
− −
where a is a germ at 0 of a real-analytic function. Since H is not transversal to M′ at
p′, it follows that a(0) = 0. Let v := f (0) for j = 1,...,n. By assumption, v ,...,v
j zj 1 n
are linearly independent vectors in Cn′. We set w = τ = 0 in (3.9), apply ∂2/∂z ∂χ and
j l
evaluate at z = χ = 0 to obtain
(3.10) v∗Av = 0, 1 j,l n,
l j ≤ ≤
where A is the n′ n′ hermitian matrix (Q′ (0)), the vectors v are regarded as n′ 1
× zα′χ′β j ×
matrices, and ∗ denotes the transpose conjugate. Note that A represents the Levi form of
M′ atp′ = H(p). Wedenotethenumber ofpositive, negative, andzeroeigenvalues ofAby
e ,e , and e , respectively, and observe that min(e ,e ) = e(M′,p′) and e = e (M′,p′).
+ − 0 + − 0 0
Let E be the n-dimensional subspace of Cn′ spanned by v ,...,v . Let : Cn′ Cn′ C
1 n
L × →
be the hermitian form given by (u,v) v∗Au. Equation (3.10) implies that restricted
7→ L
to E E is identically zero. Standard linear algebra gives n = dimE min(e ,e )+e =
+ − 0
e(M′×,p′)+e (M′,p′), contradicting (3.8). This completes the proofof≤Proposition3.3. (cid:3)
0
For the proof of Theorem 1.1, we shall also need the following proposition.
Proposition 3.4. Let M Cn+1, and M′ Cn′+1 be connected real-analytic hypersur-
faces and U an open neighb⊂orhood of M in C⊂n+1. Suppose that
(3.11) n′ +e (M′,p′) = 2n, p′ M′,
0
∀ ∈
holds. Then if H : U Cn′+1 is a holomorphic mapping with H(M) M′ such that for
→ ⊂
every p M outside a proper real-analytic subset the restriction of H to the Segre variety
∈
of M at p has rank n at p, then one of the following mutually exclusive conditions must
hold.
(i) There is an open subset V U with M V and H(V) M′.
⊂ ⊂ ⊂
(ii) H is transversal to M′ at H(p) for all p M outside a proper real-analytic subset.
∈
Proof of Proposition 3.4 . We assume, in order to reach a contradiction, that neither (i)
nor (ii) holds. Choose p M such that the restriction of H to Σ has rank n at p . By
0 ∈ p0 0
Lemma 2.1, there exists an integer k 2 such that
≥
(3.12) ρ′ H = aρk,
◦
where a is not identically zero on M. By moving to a nearby point, if necessary, we may
assume that a(p ) = 0. We choose normal coordinates (z,w) Cn C and (z′,w′)
0
Cn′ C for M and6 M′, vanishing at p and H(p ), respective∈ly. W×e write H(z,w) =∈
0 0
×
(f(z,w),g(z,w)) with f = (f1,...,fn′). As above, the defining equations of M and M′
TRANSVERSALITY OF HOLOMORPHIC MAPPINGS BETWEEN REAL HYPERSURFACES 9
can be written as w = Q(z,z¯,w¯) and w′ = Q′(z′,z¯′,w¯′), respectively. It follows from
(3.12) that
(3.13) g(z,w) Q′(f(z,w),f¯(χ,τ),g¯(χ,τ)) = a(z,w,χ,τ)(w Q(z,χ,τ))k,
− −
with a(0) = 0. Let v := f (0) for j = 1,...,n. By assumption, v ,...,v are linearly
6 j zj 1 n
indepedent vectors in Cn′. As in the proof of Proposition 3.3, we obtain (3.10), where
A is as in that proof. We denote the number of zero eigenvalues of A by e and observe
0
that e = e (M′,p′). We introduce the subspaces E,F Cn′ spanned by v ,...,v and
0 0 0 ⊂ 1 n
Av ,...,Av , respectively. Observe that the dimension of F = AE is at least n e .
1 n 0
−
By equation (3.10), it follows that E and F are orthogonal with respect to the standard
hermitian inner product of Cn′ and, hence, E F = 0 . Since n′+e = 2n, we conclude
0
that Cn′ = E F (and hence the dimension of∩F is n{ e} ). Let us denote by v := f (0)
0 w
Cn′. By settin⊕g χ = 0, τ = 0 in (3.13), we conclude t−hat g(z,w) = a(z,w,0,0)wk, and i∈n
particular, g (0) = 0, since k 2. By setting z = 0, τ = 0 in (3.13), applying ∂2/∂w∂χ ,
w j
≥
for j = 1,...,n, and evaluating at 0, we obtain v∗Av = (Av )∗v = 0. Consequently, v is
j j
orthogonal to F and, hence, v E. We set z = χ = 0 in (3.13), apply ∂k/∂wk−1∂τ, and
∈
evaluate at 0. Since a(0) = 0, we conclude that
6
∂k−1
(3.14) Q′ (f(0,w),0,0) v¯ = 0.
∂wk−1 χ′ 6
(cid:18) (cid:19)(cid:12)w=0
(cid:12)
Similarly, setting z = 0, τ = 0 in (3.13), applying ∂(cid:12)k/∂wk−1∂χ , for j = 1,...,n, and
(cid:12) j
evaluating at 0, we obtain
∂k−1
(3.15) Q′ (f(0,w),0,0) v¯ = 0, j = 1,...,n.
∂wk−1 χ′ j
(cid:18) (cid:19)(cid:12)w=0
(cid:12)
Since v E, (3.15) contradicts (3.14), compl(cid:12)eting the proof of Proposition 3.4. (cid:3)
∈ (cid:12)
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1. In view of Proposition 3.1, we may assume that (iii) of that propo-
sition holds. If condition (1.2) holds, then conclusion (ii) of Theorem 1.1 follows from
Proposition 3.3. Thus, to complete the proof, we may assume that condition (1.3) holds.
Note that if n′ + e (M′,p′) < 2n, for some p′ M′, then n′ + e (M′,p′) < 2n holds
0 0 0 ∈ 0
for all p′ in an open neighborhood V of p′ in M′ since p′ e (M′,p′) is lower semicon-
0 → 0
tinuous. Moreover, condition (1.2) then holds for all p′ V. Indeed, since e(M′,p′)
∈ ≤
(n′ e (M′,p′))/2, it follows that e(M′,p′) + e (M′,p′) (n′ + e (M′,p′))/2 < n and,
0 0 0
− ≤
hence, (1.2) holds in V. The conclusion of Theorem 1.1 follows from Proposition 3.3
(applied to V) and Remark 1.2. To complete the proof under condition (1.3), we may
assume that n′ + e (M′,p′) = 2n for all p′ M′. The conclusion of the theorem now
0
∈ (cid:3)
follows from Proposition 3.4.
10 M.S. BAOUENDI,P. EBENFELT, ANDL. P. ROTHSCHILD
4. Proofs of Theorems 1.3 and Theorem 1.4
We now give the proofs of Theorems 1.3 and Theorem 1.4.
Proof of Theorem 1.3. We suppose, in order to reach a contradiction, that (i) and (ii) of
Theorem 1.3 both fail. Hence, in view of Remark 1.2, we may assume that H is nowhere
transversal. Let p be any point on M at which e (M,p) is minimal. Note that e (M,p) is
0 0 0
then constant in an open neighborhood of p . Let us, for brevity, denote e (M,p ) by e .
0 0 0 0
Let ρ and ρ′ be local defining functions for M and M′ near p and H(p ), respectively. We
0 0
conclude, by Lemma 2.1, that there exists an integer k 2 and a real-analytic function
≥
a defined in a neighborhood of p , not divisible by ρ, such that
0
(4.1) ρ′ H = aρk.
◦
Since a 0 on M, we may assume, by moving to a nearby point if necessary, that
a(p ) = 06≡. We choose normal coordinates (z,w) Cn C and (z′,w′) Cn′ C for M
0
6 ∈ × ∈ ×
and M′, vanishing at p and H(p ), respectively, and write H(z,w) = (f(z,w),g(z,w))
0 0
with f = (f1,...,fn′). The defining equation of M′ can be written as
(4.2) 2iρ′(z′,w′,z¯′,w¯′) = w w¯ 2i z′,z¯′ ,
− − h i
where , is a nondegenerate hermitian form, and that of M as
h· ·i
n−e0
(4.3) 2iρ(z,w,z¯,w¯) = w w¯ 2i δ z z¯ +ψ(z,z¯,w+w¯),
j j j
− −
j=1
X
where ψ(z,0,s) = ψ(0,z¯,s) = 0, ψ(z,z¯,s) = O(3), and δ = 1. Now, identity (4.1)
j
±
becomes
¯
(4.4) g(z,w) g¯(χ,τ) 2i f(z,w),f(χ,τ) =
− − h i
n−e0 k
b(z,w,χ,τ) w τ 2i δ z χ +O(3) ,
j j j
− −
(cid:18) j=1 (cid:19)
X
where b is a holomorphic function defined in a neighborhood of 0 in C2n+2, with b(0) = 0.
We introduce the following vectors in Cn′ 6
(4.5) vj := fzj(0), uj := fzjk−2w(0), xj := fzjwk−1(0), j = 1,...,n−e0,
where the subscripts denote partial derivatives. By carefully identifying appropriate
monomials on both sides in (4.4), we conclude that the following identity of 3(n e )
0
− ×
